MRI Spring School 2004
Lie Groups in Analysis, Geometry and Mechanics
Contents of the Courses
Structure Theory of Lie Groups and Lie Algebras
by J.A.C. Kolk.
In this course we shall treat results that are foundational for the other courses. We begin with a review of the structure of compact semisimple Lie groups
and Lie algebras, with attention to maximal tori, root space decompositions, Weyl groups and the Weyl integration formula. Here we shall see a strong
interaction with the course Group Actions. We also study examples of noncompact Lie groups, in particular, SL(n,R), and decompositions
of such groups.
Next comes the abstract representation theory of compact Lie groups and the classification by highest weight of the finite-dimensional representations.
The algebraic tools needed for this, the universal enveloping algebra and the Poincaré-Birkhoff-Witt Theorem, will also be studied, as well as the
related theory of Verma modules.
Literature
J.J. Duistermaat and J.A.C. Kolk:
Lie Groups, Springer-Verlag, Berlin,
2000
J. Adams and D. Vogan, eds.:
Representation Theory of Lie Groups,
IAS/Park City Mathematics Series, Volume 8, American Mathematical Society, Providence, 2000
Group Actions
by J.A.C. Kolk.
For smooth actions of a compact Lie group on a smooth manifold,
or slightly more generally for proper Lie group actions,
very detailed descriptions can be given of the orbits,
the action in a suitable invariant neighborhood of an orbit,
the decomposition of the manifold into orbit types,
the space of invariant smooth functions, and various
other aspects of the action.
The local action, in a suitable invariant neighborhood of any given point,
is isomorphic to the action on associated vector bundles,
which is described entirely in terms of the Lie group itself,
the stabilisator subgroup of the point, and a linear representation
of this stabilisator subgroup. It is a theorem of Schwarz that the
algebra of smooth functions which are invariant under such a linear
representation is generated by finitely many homogeneous
polynomials. This leads to a local description of
the orbit space as a semi-algebraic variety with a conic
singularity. These features of actions
will be used in the course Symmetry in Mechanics.
Particular examples are the actions by conjugation of a compact Lie
group on itself and on its Lie algebra.
The theory then leads to the
structure theory of compact Lie groups and their Lie algebras,
including the description of the maximal tori, roots and root spaces,
the Weyl group and Weyl's integration theorem. See also the
course Structure Theory of Lie Groups and Lie Algebras.
Another example arises when the stabilisator group is trivial.
When the orbit space is smooth we have a principal
fiber bundle, which appears in the course Analysis on Principal Fiber Bundles.
There is a close relation between proper smooth actions and
algebraic actions of reductive complex algebraic groups on
complex affine varieties, but there will probably not be
enough time to work this out in detail.
Prerequisites
The basics of differential geometry, at least
a working knowledge of manifolds, tangent spaces and tangent mappings.
Some familiarity with fiber bundles.
Symplectic Geometry
by J.J. Duistermaat
By definition, a symplectic form on a finite-dimensional vector
space E is a non-degenerate anti-symmetric bilinear form
$\sigma$ on E. One may be more familiar with a bilinear form
that is symmetric; when positive definite, such
a form is an inner product. Just as for inner products,
orthogonal complements can be taken with
respect to the symplectic form $\sigma$. However, a novel feature
is that there are lots of linear subspaces, called Lagrange
subspaces, that are equal to their own orthogonal complements.
These Lagrange subspaces can be used to find a basis on
which $\sigma$ has a standard form. We will make some
remarks on the manifold of all Lagrange planes and on the
symplectic group, the group of all linear transformations
in E which leave $\sigma$ invariant.
A symplectic form on a smooth manifold M is a closed smooth
differential form $\sigma$ of degree two on M such that,
for each m in M, the bilinear form $\sigma_m$ is non-degenerate.
Thus, $\sigma_m$ is a symplectic form on the tangent space T_mM
of M at each point m. The previous analogy with inner products
makes that we may think of $\sigma$ as an anti-symmetric analogue of
a Riemannian structure on M. The closedness of $\sigma$
is analogous to the condition that the Riemannian structure is flat.
In fact, the closedness condition implies the Darboux lemma,
which says that, locally, any symplectic form can be brought
into the standard form discussed above. As there are lots
of Lagrange planes, there are also lots of Lagrange manifolds;
these are defined as the smooth submanifolds L of M such that
for each m in L the tangent space T_mL
is a Lagrange subspace of the tangent space T_mM,
with respect to the symplectic form $\sigma_m$.
Although Riemannian geometry is more well-known than symplectic
geometry, the latter has a wealth of applications, as it is
one of the basic structures of classical (Hamiltonian) mechanics, general
non-linear first order partial differential equations (Hamilton-Jacobi)
theory, and complex projective algebraic geometry.
We plan to discuss some these applications at least to the extend
that the basic role of symplectic differential geometry
becomes apparent. The understanding of symplectic structures will
be useful in the courses Symmetry in Mechanics and The Momentum Map.
Prerequisites
The basics of differential geometry, at least
a working knowledge of manifolds, tangent spaces and tangent mappings,
as well as differential forms.
Symmetry in Mechanics
by R.H. Cushman.
This course will treat the theory of symmetry in Hamiltonian
systems.
The basic notion of a momentum mapping for a
Hamiltonian action of a Lie group on a symplectic manifold will be
defined and used throughout the course. When the Hamiltonian action is proper,
we will develop
the theory of reduction of symmetries, which lowers the
dimension of the space where the motion takes place.
Both the regular case, when the action is
free, and the singular case, when it is not, will be treated.
The theory will be illustrated by a thorough study of the following
systems: the two dimensional harmonic oscillator,
the Euler top, the spherical pendulum and the Lagrange top.
Particular attention will be paid to understanding the global geometry
of the momentum mappings of these integrable systems.
Prerequisites
A basic knowledge of:
1) Symplectic geometry; in particular, what a symplectic
manifold, a symplectic form, a Hamiltonian vector field, and
Poisson brackets are;
2) Action of a Lie group on a manifold. Some
familiarity with classical mechanics would be useful.
Representation Theory and
Applications in Classical
Quantum
Mechanics
by E.P. van den Ban.
In quantum mechanics, the state space of a system (like a particle) is the space
of rays in a Hilbert
space H. In the context of
special relativity, the coordinate systems of inertial observers are related by
transformations from
the Poincaré group P, i.e.,
the semi-direct product of the Lorentz group L and the group of translations
of Minkovski space. If
the system is invariant
under these transformations, a theorem of E. Wigner asserts that associated
transformations in state
space
are given by a unitary representation of P (or rather its double cover) in the
Hilbert space H.
The classification of the irreducible
unitary representations of P (or rather its double cover) is therefore of
utmost importance for the
classification
of particles.
The classification of the physical relevant irreducible
unitary representations of P was achieved by Wigner in 1939,
the full classification by V. Bargmann in 1947.
Both authors used the method of induction, by which
one obtains representations of a Lie group G from representations
of a subgroup H. These works generated a lot of interest
in the representation theory of non-compact Lie groups.
We will discuss several aspects of the theory of representations
that eventually evolved, with the Poincaré group as
our guiding example.
Subjects that will be discussed are: induced representations,
Mackey's imprimitivity theorem, Heisenberg group, Stone-von Neumann theorem,
highest weight theory and Borel-Weil description
of representations of compact groups,
induced representations of non-compact semisimple groups, Harish-Chandra
modules, representations of the principal and the discrete series
of SL(2,R).
Analysis on Principal Fiber Bundles
by J.J. Duistermaat.
Often a vector bundle V over a smooth manifold M can be described as
an associated vector bundle, which means that there is a
principal fiber bundle P over M with structure group G
and a representation $\rho$ of G in a finite-dimensional complex
vector space E, in such a way that the space $\Gamma$
of smooth sections of V can be identified with the space
of G-equivariant smooth mappings from P to E. The analysis
of linear partial differential operators acting on $\Gamma$ is
greatly helped by studying the corresponding operators
on the space of E-valued functions on P, which is
simpler to deal with than the space of sections of V.
This point of view is important for the theory of induced representations,
which will be discussed in the course Representation Theory.
If M is provided with a Riemannian structure and V with a
Hermitian connection, then one has a naturally defined Laplace
operator $\Delta^M$ which acts on $\Gamma$.
Using a G-invariant connection on the principal fiber
bundle $P\to M$ and a conjugacy invariant inner product on the
Lie algebra L(G) of G, we obtain a Riemannian structure on P
and a corresponding Laplace operator $\Delta^P$
acting on E-valued functions on P. Adding a zero order
term C which comes from the Casimir operator of L(G),
one obtains an operator $\Delta^P+C$ such that
$\Delta^M$ is equal to the restriction to $\Gamma$ of the
operator $\Delta^P+C$. For short time asymptotics of the
corresponding heat kernels, the geodesics in M and P
play an important role, whereas for the geodesics on P the derivative of the exponential mapping $:L(G)\to G$ enters
into the formulas. This leads to the explanation - by
Berline and Vergne - of the appearance of the Todd class
in the formula for the index of for instance the spin-c Dirac operator.
The Todd class and the easier Chern classes are examples of
characteristic classes, which can be introduced by means
of equivariant cohomology. The latter can be viewed as another
example of analysis on principal fiber bundles.
Prerequisites
Basics of differential geometry, Lie groups and linear partial differential
equations.
The Momentum Map from an Algebraic and Differential
Geometric Point of
View
by
G.J. Heckman.
If a Lie group K acts on a symplectic manifold M in a hamiltonian way
(a slightly stronger condition than preservation of the symplectic form)
then this yields a momentum map J from M to the dual of the Lie algebra
of K. We shall focus on the case that K is compact and J is a proper
map.
There are two basic cases to keep in mind. In the first case M is the
cotangent bundle (the phase space) of some manifold (the configuration space)
and the action of the group comes from an action on configuration space.
Secondly, M is a complex projective manifold with a hermitian metric
induced from a Fubini-Study metric invariant under the action of the group.
The first case
is the setting of classical mechanics with a symmetry group;
quantization of such a situation leads to questions of representation
theory of K and spectral theory of differental operators commuting with (spectral degeneration). In the second case one is lead to complex
algebraic geometry and geometric invariant theory.
The main point I would like to explain is the remarkable analogy between
these two at first sight rather different kinds of examples.
Some concepts that will be discussed are: reduced phase space, reduced
hamiltonian, convexity theorem, variation of reduced
phase space, [Q,R]=0.
Literature
R. Abraham and J. Marsden, Foundations of Mechanics
V. Guillemin and S. Sternberg, Symplectic Techniques in Physics
D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory